In the setting of [1]}, we consider classical (i.e. deterministic) strategies: the provers' strategy is described
by a “points function,” assigning a value in \(\Sigma \)  to each point
\(u \in [n]^m\) , and a
“lines function,” assigning a codeword of \(\mathcal {C}\)  to each
axis-parallel line \(\ell \subset [n]^{m}\)  queried in the test. From the assumption that the
points and lines functions agree at a randomly chosen point with high
probability, we would like to construct a “global” codeword of \(\mathcal {C}^{\otimes m}\)  that has high agreement with
the “local” points function, on average over a uniformly random \(u\)  at which both are evaluated. This is done by inductively constructing
“subspace functions” defined on axis-aligned “affine subspaces”: subsets of \([n]^m\)  where \(m-k\)  coordinates have been fixed, for some \(1 \le k \le m\) . We refer to the parameter \(k\)  as the dimension of the subspace. As special cases, a subspace of dimension \(k = 0\)  is a single point in \([n]^{m}\) , a subspace of dimension \(k =1\)  is an axis-aligned line, and a subspace with dimension \(k = m\)  is the entire space \([n]^m\) .
